CHAPTER VI.

THE INVERSE PROBLEM.

534. Nature of the Inverse Problem.—By the inverse problem is here meant a certain problem so-called by Jevons. Its nature will be indicated by the following extracts, which are from the Principles of Science and the Studies in Deductive Logic respectively.

“In the Indirect process of Inference we found that from certain propositions we could infallibly determine the combinations of terms agreeing with those premisses. The inductive problem is just the inverse. Having given certain combinations of terms, we need to ascertain the propositions with which they are consistent, and from which they may have proceeded. Now if the reader contemplates the following combinations,—

ABCabC
aBCabc,

he will probably remember at once that they belong to the premisses A = AB, B = BC. If not, he will require a few trials before he meets with the right answer, and every trial will consist in assuming certain laws and observing whether the deduced results agree with the data. To test the facility with which he can solve this inductive problem, let him casually strike out any of the possible combinations involving three terms, and say what laws the remaining combinations obey. Let him say, for instance, what laws are embodied in the combinations,—

ABCaBC
AbcabC,

“The difficulty becomes much greater when more terms enter 526 into the combinations. It would be no easy matter to point out the complete conditions fulfilled in the combinations,—

ACe
aBCe
aBcdE
abCe
abcE.

After some trouble the reader may discover that the principal laws are C = e, and A = Ae ; but he would hardly discover the remaining law, namely that BD = BDe” (Principles of Science, 1st ed., vol. I., p. 144; 2nd ed., p. 125).

“The inverse problem is always tentative, and consists in inventing laws, and trying whether their results agree with those before us” (Studies in Deductive Logic, p. 252).

The problem may preferably be stated as follows:—
Given a complex proposition of the form

Everything is P1P2 … or Q1Q2 … or …,

to find a set of propositions not involving any alternative combination of terms, which shall together be equivalent to it.[520]

[520] The problem may also be stated as follows:—Given a universal affirmative complex proposition containing alternative terms to find an equivalent compound conjunctive proposition all the determinants of which are affirmative and free from alternative terms.

It may be observed that Jevons does not definitely exclude alternative terms in his solutions of inverse problems, though he generally seeks to avoid them. The problem cannot, however, be defined with accuracy unless such terms are explicitly excluded.

The inverse problem is in a sense indeterminate, for we may find a number of sets of propositions, not involving any alternative combination of terms, which are precisely equivalent in logical force, and hence any inverse problem may admit of a number of solutions. But it is not necessary to have recourse to a series of guesses in order to solve any inverse problem, nor need the method of solution be described as wholly tentative. Several systematic methods of solution applicable to any inverse problem are formulated in the following sections. Since, however, more solutions than one are possible, some of which are simpler than others, the process may be regarded as more or less tentative in so far as we seek to obtain the most satisfactory solution.

The following may be taken as our criterion of simplicity. Comparing two equivalent sets of propositions, not involving any 527 alternative combination of terms, that set may be regarded as the simpler which contains the smaller number of propositions. If each set contains the same number of propositions, then we may count the number of terms involved in their subjects and predicates taken together, and regard that one as the simpler which involves the fewer terms.

535. A General Solution of the Inverse Problem.—Let us suppose, then, that we are given a complex proposition involving alternative combination, and that we are to find a set of propositions, not involving alternative combination, which shall together be equivalent to it.

The data may be written in the form

Everything is P or Q or S or T or &c.,

where P, Q, &c., are themselves complex terms involving conjunctive, but not alternative, combination.[521]

[521] The proposition in its original form may admit of simplification in accordance with the rules laid down in [chapter 1]. It will generally speaking be found advantageous to have recourse to such simplification before proceeding further with the solution.

By contraposition one or more of these complex terms may be brought over from the predicate into the subject, so that we have

Whatever is not either P or S or &c. is Q or T or &c.

The selection of certain terms for transposition in this way is arbitrary (and it is here that the indeterminateness of the problem becomes apparent); but it will generally be found best to take two or three which have as many common determinants as possible.

What is not either P or S or &c. is Q or T or &c.

will, when the subject is written in the affirmative form, be immediately resolvable into a series of propositions, which taken together give all the information originally given.[522] Any of these propositions which still involve alternative combination may be dealt with in the same way, until no alternative combination remains.

[522] See section [446].

We shall now be left with a set of propositions which will satisfy the required conditions. The possibility of various simplifications has, however, to be considered. Thus, it will be necessary to make sure that each of the propositions is itself expressed in its simplest form;[523] and to observe whether any two or more of the propositions 528 admit of a simple recombination.[524] It may also be found that some of the propositions can be altogether omitted, inasmuch as they add nothing to the information jointly afforded by the remainder; or that, considered in their relation to the remaining propositions, they may, at any rate, be simplified by the omission of one or more of the terms which they contain.[525] When these simplifications have been carried as far as is possible we shall have our final solution.[526]

[523] For example, All AB is BC may be reduced to All AB is C.

[524] For example, All ac is d and All Bc is d may be combined into All cD is Ab.

[525] Thus, for the propositions All AB is CD and All Ab is C we may substitute the propositions All AB is D and All A is C.

[526] It may be observed that it is no part of our object to obtain a set of propositions which are mutually independent. As a matter of fact, it will generally be found that the maximum simplification involves the repetition of some items of information. Thus, in the example given in the preceding note the propositions All AB is CD and All Ab is C are quite independent of one another; but the proposition All A is C renders superfluous part of the information given by the proposition All AB is D.

The solution may, if we wish, be verified by recombining into a single complex proposition the propositions that have been obtained, an operation by which we shall arrive again at a series of alternants substantially identical with those originally given us. Such verification is, however, not essential to the validity of our process, which, if it has been correctly performed, contains no possible source of error.

The following examples will serve to illustrate the above method.

I. For our first example we may take one of those chosen by Jevons in the extract quoted in the preceding section.

Given the proposition, Everything is either ABC or Abc or aBC or abC, we are to find a set of propositions not involving alternative combination which shall be equivalent to it.
By the reduction of aBC or abC to aC, followed by contraposition, we have What is neither ABC nor Abc is aC ; therefore, What is a or Bc or bC is aC ; and this may be resolved into the three propositions:—

⎧All a is C,
⎨Bc is non-existent,
⎩All bC is a.

Bc is non-existent is reducible to All B is C ; and this proposition and All a is C may be combined into All c is Ab.

529 Hence we have for our solution the two propositions:—

⎰All c is Ab,
⎱All bC is a.

It will be found that by the recombination of these propositions we regain the original proposition.

II. We may next take the more complex example contained in the same extract from Jevons.

The given alternants are ACe, aBCe, aBcdE, abCe, abcE ; and by the reduction of dual terms, they become aBcdE, abcE, Ce. Therefore, What is not aBcdE or abcE is Ce ; and this proposition may be resolved into the four propositions:—

⎧
⎨
⎩		All A is Ce ;(1)
All BD is Ce ;(2)
All C is e ;(3)
All e is C.(4)

But since by (3) All C is e, (1) may be reduced to All A is C ; and this proposition may be combined with (4) yielding All c is aE. Also by (3), (2) may be reduced to All BD is C.

Hence our solution becomes

⎧All BD is C,
⎨All C is e,
⎩All c is aE.

This solution may be shewn to be equivalent to the solution given by Jevons himself.

III. The following problem is from Jevons, Principles of Science, 2nd ed., p. 127 (Problem v).

The given alternants are ABCD, ABCd, ABcd, AbCD, AbcD, aBCD, aBcD, aBcd, abCd.

By the reduction of duals these alternants may be written as follows: ABC or ABcd or AbD or aBCD or aBc or abCd.

Therefore, by contraposition, Whatever is not ABC or AbD or aBc is ABcd or aBCD or abCd.

But Whatever is not ABC or AbD or aBc is equivalent to Whatever is ABc or aBC or ab or bd. Hence we have for our solution the following set of propositions:

(1)  All ABc is d,     (2)  All aBC is D,
(3)  All ab is Cd,     (4)  All bd is a.[527]

This is equivalent to the solution given by Jevons, Studies, p. 256.

[527] We first obtain All bd is aC ; but since by (3) All abd is C, this may be reduced to All bd is a.

530 IV. The following example is also from Jevons, Principles of Science, 2nd edition, p. 127 (Problem viii). In his Studies, p. 256, he speaks of the solution as unknown. A fairly simple solution may, however, be obtained by the application of the general rule formulated in this section.

The given alternants are ABCDE, ABCDe, ABCde, ABcde, AbCDE, AbcdE, Abcde, aBCDe, aBCde, aBcDe, abCDe, abCdE, abcDe, abcdE.

By the reduction of duals these alternants may be written: ABCe or ABcde or Abcd or ACDE or aBCde or abdE or aDe.

Therefore, by contraposition, Whatever is not either ABCe or ABcde or Abcd or abdE or aDe is ACDE or aBCde.

But it will be found that, by the application of the ordinary rule for obtaining the contradictory of a given term, Whatever is not either ABCe or ABcde or Abcd or abdE or aDe is equivalent to Whatever is AbC or ade or BE or AcD or DE.

Hence our proposition is resolvable into the following:

 (i)  All AbC is DE ;
(ii)  All ade is BC ;
(iii)  All BE is ACD ;
(iv)  AcD is non-existent ;
 (v)  All DE is AC.

But by (v) All BE is AC or d ; therefore, (iii) may be reduced to All BE is D. Again by (iv), All DE is a or C ; therefore, (v) may be reduced to All DE is A.

Hence we have the following as our final solution:—

(1)  All AbC is DE ;
(2)  All ade is BC ;
(3)  All BE is D ;
(4)  All cD is a ;
(5)  All DE is A.

536. Another Method of Solution of the Inverse Problem.—Another method of solving the inverse problem, suggested to me by Dr Venn, is to write down the original complex proposition in the negative form, i.e., to obvert it, before resolving it. It has been already shewn that a negative proposition with an alternative predicate may be immediately broken up into a set of simpler propositions.

In some cases, especially where the number of destroyed combinations as compared with those that are saved is small this plan is of easier application than that given in the preceding section.

531 To illustrate this method we may take two or three of the examples already discussed.

I. Everything is ABC or Abc or aBC or abC ;
therefore, by obversion, Nothing is AbC or ac or Bc ;
and this proposition is at once resolvable into

⎰All Ab is c,
⎱All c is Ab.[528]

[528] The equivalence between this and our former solution is immediately obvious. Equationally it would be written Ab = c.

II. Everything is ACe or aBCe or aBcdE or abCe or abcE ; therefore, by obversion, Nothing is Ac or BcD or CE or ce.

This proposition may be successively resolved as follows:

⎧No c is A or e,
⎨No E is C,
⎩No BD is c.
⎧All c is aE,
⎨All E is c,
⎩All BD is C.

III. Everything is ABCD or ABCd or ABcd or AbCD or AbcD or aBCD or aBcD or aBcd or abCd ; therefore, by obversion, Nothing is ABcD or Abd or aBCd or abc or abD ; and this proposition may be successively resolved as follows:

⎧
⎨
⎩		No ABc is D ;
No bd is A ;
No aBC is d ;
No ab is c or D.
⎧
⎨
⎩		All ABc is d ;
All bd is a ;
All aBC is D ;
All ab is Cd.

It is rather interesting to find that notwithstanding the indeterminateness of the problem we obtain by independent methods the same result in each of the above cases.

537. A Third Method of Solution of the Inverse Problem.—The following is a third independent method of solution of the inverse problem, and it is in some cases easier of application than either of the two preceding methods.

532 Any proposition of the form

Everything is ……

may be resolved into the two propositions:

⎰All A is ……
⎱All a is ……

which taken together are equivalent to it; similarly All A is …… may be resolved into the two All AB is ……, All Ab is …… and it is clear that by taking pairs of contradictories in this way we may resolve any given complex proposition into a set of propositions containing no alternative terms. Redundancies must of course as before be as far as possible avoided.

To illustrate this method we may again take the first three examples given in section 535.

I. Everything is ABC or Abc or aBC or abC may be resolved successively as follows:

⎰All C is AB or aB or ab ;
⎱All c is Ab.
⎰All bC is a ;[529]
⎱All c is Ab.

[529] Taking BC as our subject we have All BC is A or a, and since this is a merely formal proposition, it may be omitted.

II. Everything is ACe or aBCe or aBcdE or abCe or abcE may be resolved successively as follows:

⎰All C is Ae or aBe or abe ;
⎱All c is aBdE or abE.
⎧All C is e ;
⎨All c is aE ;
⎩All c is Bd or b.
⎧All C is e ;
⎨All c is aE ;
⎩All Bc is d.

III. Everything is ABCD or ABCd or ABcd or AbCD or AbcD or aBCD or aBcD or aBcd or abCd may be resolved successively as follows:

⎰All B is ACD or ACd or Acd or aCD or acD or acd ;
⎱All b is ACD or AcD or aCd.
⎰All B is AC or aD or cd ;
⎱All b is AD or aCd.  533
⎧
⎨
⎩		All BC is A or aD ;
All Bc is aD or d ;
All Ab is D ;
All ab is Cd.
⎧
⎨
⎩		All BCd is A ;
All ABc is d ;
All Ab is D ;
All ab is Cd.

The above solutions are practically the same as those obtained in the two preceding sections.

538. Mr Johnson’s Notation for the Solution of Logical Problems.—In his articles on the Logical Calculus Mr Johnson proposes a notation by the aid of which the solution of inverse problems may be facilitated. It consists in representing conjunctive combination by horizontal juxtaposition, and alternative combination by vertical juxtaposition. A bar—drawn horizontally or vertically—serves the purpose of a bracket where necessary. Thus,

represents AB or CD;

represents (A or C) and (B or D). These two forms are of course not equivalent to each other. But if contradictories are placed in a pair of diagonally opposite corners, then the combination is the same in whichever way we read it. Thus,

represents AB or aC ;

represents (A or C) and (a or B). But these are equivalent to each other; for (A or C) and (a or B) is equivalent to AB or aC or BC, and—since BC by development is ABC or aBC—this is equivalent to AB or aC. Mr Johnson continues as follows:—“By adopting the plan of placing successive letter-symbols in opposite corners we may solve the inverse problem with surprising ease. The method of solution closely resembles the third of those adopted by Dr Keynes, and it was this that suggested mine. I will, therefore, illustrate by taking Dr Keynes’s three examples which are the following:—

534 Here the columns or determinants may be read off:—

(C or Ab) and (B or a or c) = (If c, then Ab) and (If AC, then B).

This is read: (If c, then aE) and (If BD, then C) and (If C, then e).

That is: (If ab, then Cd) and (If bd, then a) and (If ABD, then C) and (If BCd, then A). In this last problem, we first place B and b opposite; then for the B alternants, we place C and c opposite, and for the b alternants A and a. To get the simplest result, we should aim at dividing the columns into as equal divisions as possible.

The notation thus explained enables us to solve any problems in a simple manner. The expression in its final form may be read equally well in columns or in rows, i.e., as a determinative or as an alternative synthesis. Of course, a precisely similar process may be used, if we started with determinatively given or mixed data” (Mind, 1892, p. 351).

539. The Inverse Problem and Schröder’s Law of Reciprocal Equivalences.—The inverse problem may also be solved, though somewhat laboriously, by the aid of the reciprocal relation between the laws of distribution given in section [428], this reciprocal relation depending upon the law that to every equivalence there corresponds another equivalence in which conjunctive combination is throughout substituted for alternative combination and vice versâ. Thus, by the first law of distribution, (A or B) and (C or D) = AC or AD or BC or BD, and hence follows the corresponding equivalence AB or CD = (A or C) and (A or D) and (B or C) and (B or D). In this way any inverse problem may be practically resolved into the more 535 familiar problem of conjunctively combining a series of alternative terms.[530]

[530] It will be observed that the inverse problem involves the transformation of a logical expression consisting of a series of alternants into an equivalent expression consisting of a series of determinants. Schröder’s Law of Reciprocity shews that the process required for this transformation is practically the same as that by which an expression consisting of a series of determinants is transformed into an equivalent expression consisting of a series of alternants.

Taking as an example the first problem given in section [535], we may proceed as follows: (A or B or C) and (A or b or c) and (a or B or C) and (a or b or C) = (A or Bc or bC) and (a or C) = AC or aBc or bC. Therefore, we have the corresponding equivalence ABC or Abc or aBC or abC = (A or C) and (a or B or c) and (b or C). Hence the proposition Everything is ABC or Abc or aBC or abC may be resolved into the three propositions, Everything is A or C, Everything is a or B or c, Everything is b or C ; and we have for our solution of the inverse problem: All c is A, All bC is a, All c is b ; or, combining the first and last of these propositions, All c is Ab, All bC is a.

Similarly, the second problem in section [535] may be solved as follows:—(A or C or e) (a or B or C or e) (a or B or c or d or E) (a or b or C or e) (a or b or c or E) = aC or bCd or CE or ce. Hence the corresponding equivalence ACe or aBCe or aBcdE or abCe or abcE = (a or C) (b or C or d) (C or E) (c or e); and we have for our solution of the inverse problem, All A is C, All BD is C, All c is E, All C is e ; or, combining the first and third of these propositions, All c is aE, All BD is C, All C is e.

EXERCISES.

540. Find propositions that leave only the following combinations, ABCD, ABcD, AbCd, aBCd, abcd. [Jevons, Studies, p. 254.]

Jevons gives this as the most difficult of his series of inverse problems involving four terms. It may be solved as follows:—
Everything is ABCD or ABcD or AbCd or aBCd or abcd ; therefore, by contraposition and the reduction of dual terms, Whatever is not either AbCd or aBCd is ABD or abcd.
536 Therefore, Whatever is AB or ab or c or D is ABD or abcd ; and this is resolvable into the four following propositions:

⎧
⎨
⎩		All AB is D,   (1)
All ab is cd,   (2)
All c is ABD or abd,   (3)
All D is AB.   (4)

Since by (4) All D is AB, and by (2) All ab is d, (3) may be reduced to All c is D or ab, and therefore to All cd is ab. Also, by (4) All ab is d, and hence (2) may be reduced to All ab is c.

Our set of propositions may therefore be expressed as follows:—

⎧
⎨
⎩		All AB is D,
All ab is c,
All cd is ab,
All D is AB.[531]

[531] Restoring the second of these propositions to the form All ab is cd, and writing the propositions equationally, the solution may be expressed in a still simpler form, namely, AB = D, ab = cd.

541. Resolve the proposition Everything is ABCDeF or ABcDEf or AbCDEF or AbCDeF or AbcDeF or aBCDEf or aBcDEf or abCDeF or abCdeF or abcDef or abcdef into a conjunction of relatively simple propositions.

[Jevons, Principles of Science, 2nd ed., p. 127 (Problem x.)]

The following is a solution:—

(1) All A is D ;
(2) All ABC is e ;
(3) All aF is bCe ;
(4) All Bf is DE ;
(5) All bf is ace ;
(6) All cF is be.

This is somewhat less complex than the solution by Dr John Hopkinson given in Jevons, Studies in Deductive Logic, p. 256, namely:—

(i) All d is ab ;
(ii) All b is AF or ae ;
(iii) All Af is BcDE ;
(iv) All E is Bf or AbCDF ;
(v) All Be is ACDF ;
(vi) All abc is ef ;
(vii) All abef is c.

537 542. How many and what non-disjunctive propositions are equivalent to the statement that “What is either Ab or bC is Cd or cD, and vice versâ”? [Jevons, Studies, p. 246.]

The given statement is at once resolvable into the four following propositions:

⎧
⎨
⎩		All Ab is Cd or cD, (i)
All bC is Cd or cD, (ii)
All Cd is Ab or bC, (iii)
All cD is Ab or bC. (iv)
(i) may be resolvedinto⎰All Abc is D, (v)
⎱All AbD is c. (vi)

But (vi) is inferable from (ii); and observing some other obvious simplifications we obtain immediately the following solution:

(1) All Abc is D ;
(2) All bC is d ;
(3) All Cd is b ;
(4) All cD is Ab.

543. Shew the equivalence between the two sets of propositions given in section [541]. [K.]

544. Find which of the following propositions may be omitted without affecting the information given by the propositions as a whole: All Ab is cDE ; All Ac is bDE ; All Ad is BCe ; All Ae is BCd ; No aE is B or C ; No B is c ; All Bd is ACe ; No bD is C or e ; No bE is Ad or C ; All C is B ; All Cd is ABe ; All cD is bE ; All cE is AbD or ab ; All de is ABC or abc. [K.]

545. Resolve each of the following complex propositions into a conjunction of propositions not containing any alternative combination of terms:

(1) Everything is ABCD or AbCd or aBcD or abcd ;
(2) Everything is AbCD or AbCd or Abcd or aBcd or abCD or abCd or abcd ;
(3) Everything is AbcDE or aBCd or aBCE or aBcd or aBde or abCe or abce or abDe or abde or BcdE or bCDe ;
(4) Everything is ABCE or ABcd or ABcE or ABde or Abcd or abCE or abcE or abdE or abde or BCde ;
(5) Everything is ABCDE or ABCdE or ABcDE or ABcDe or ABcde or AbCdE or Abcde or aBCDE or aBCde or abCDE or abcDe ;
538 (6) Everything is ABDe or ABDF or AcDe or Acef or aBDe or aBDF or abCD or abCd or abcD or abcd or aCDE or aCDe or aCdE or aCde or acDe or aDEF or aDEf or aDeF or aDef or BcDF or bceF or bcef ;
(7) Everything is AbdE or Abef or AbF or Acdef or aBDF or abCF or aCdE or ade or bCDe or bCdf or bDEF ;
(8) Everything is ABCEf or Abe or aBCdf or aBcdE or aBcdeF or abef or bceF. [K.]

546. Express the following proposition in as small a number as you can of propositions in which no alternative combination of terms occurs: Everything is ABCDe or ABCdE or ABcDe or AbCdE or AbCde or aBCdE or aBcDE or aBcde or aBcdE or abCde or abCdE. [J.]

547. Solve the fourth problem given in section [535], (α) by the method described in section 536, (β) by that described in section [537]. [K.]

548. Solve the problem given in section [540] and also the fourth problem given in section [535] by aid of the notation described in section [538]. [K.]

549. Solve the third and fourth problems given in section [535] by the method described in section [539]. [K.]

550. Shew that any universal complex proposition may be resolved into a set of propositions in which no conjunctive combination of terms occurs. [K.]